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Let RN with N 2. We consider the equations N X i=1 uai @ 2u @x2i +p(x )=0 ; u j @ =0 ; with a1 a2 :::: aN 0a nd a 1 >a N. We show that if is a convex bounded region in RN , there exists at least one classical solution to this boundary value problem. If the region is not convex, we show the existence of a weak solution. Partial results for the existence of classical solutions for non-convex domains in R2 are also given.

On a singular quasilinear anisotropic elliptic boundary value problem